Stationarity method condition

Slightly overestimated Pi as compared with the pressure calculated on the basis of the one-centered approach may be conditioned by (i) the approximation related to the application of the partial stationarity method by concentrations of chain carriers (ii) the consumption of the initial reagents when reaching the first explosion limit. 

Radical polymerizations are almost always considered as kinetically stationary. However, the stationarity conditions are not always fulfilled. Living polymerizations with rapid initiation are stationary, but the character of the medium should not significantly change during polymerization in order to prevent shifts in the equilibria between ion pairs and free ions. All other polymerizations are non-stationary even, to some extent, living polymerizations with slow initiation. It is usually very difficult to define initiation and termination rates so as to permit exact kinetic analysis. When the concentration of active centres cannot be directly determined, indirect methods must be applied, and sometimes even just a trial search for best agreement with experiment. 

The second possibility is to use a gradient code, if this is available for the chosen method. The third method is the simplest, namely to evaluate E2 as an expectation value. This method is equivalent to the other two, if Eq satisfies a stationarity condition, like the Brillouin condition of Hartree-Fock theory. For non-stationary approaches, like MP2 or CC, the methods based on differentiations of the expectation value are more reliable. This is related to the validity or non-validity of the Hellmann-Feynman theorem. 

This approximation resolves the computational difficulty encountered in the direct exact formulation that requires repeated computations of the solution of linear simultaneous algebraic equations and determinants of the matrices with huge dimensions. The efficiency in the approximated expansion is gained by the appreciation that the conditioning information can be truncated within one period of the system only. For linear systems, the expressions for the reduced-order likelihood function p(yi, yj, - - -, yNp W, C) and the conditional PDFs p(.yn 0, yn-Np, yn-Np+1, , y -i, C) are available since they are Gaussian and the correlation functions are known in closed forms regardless of the stationarity of the response. For stationary response, the method is very efficient in the sense that evaluation of all the conditional PDFs p(ynW, yn-Np,yn-Np+i,, y -i, C) requires the inverse and determinant of two relatively small matrices only. 

While this equation is quite simple in form, it hides the fact that the universal functional F[p] is not available in explicit form. Numerous schemes have been formulated for a direct optimization of the density based on these stationarity conditions, but these methods have not really been competitive. The most efficient approach has been to invoke a quasi-independent-particle approximation, formulated in the Kohn-Sham equations. 

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